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Implications 2. Dynamical quantum theories cant be simulated in BQP, relative to oracle Define joint distribution over values of observable at times t 1, t 2, etc. (I.e. classical history) Given polytime quantum algorithm and set of sampling points, how hard to sample from this distribution?

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How to Find a Collision in O(1) Queries If Your Memory Is Perfect 1.Prepare and observe 2 nd register If X is 2-1, obtain (|i +|j )/ 2 with x i =x j 2.Sample 3. Hadamard every bit, and sample again 4. Hadamard every bit again (returning to (|i +|j )/ 2), and sample again Which basis state (|i or |j ) were you in after Step 2? After Step 4?

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How to Handle n mod g 0: Sketch Choose N slightly larger than n such that g divides N Choose g-1 function on {1,…,N} u.a.r, then subfunction of size n Acceptance prob. close to bivariate polynomial in g,N for all g|N s.t.

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(continued) Restrict gs range to [1,G]; then (g,N) points with g|N are plentiful, so P is bounded P has large derivative somewhere in either the g or N directions Lower bound obtained when G=n 2/5 :

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Large derivative between 1-1 and 2-1 Lots of points at which g|N so P is bounded

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Shis Improvement to (n 1/4 ) Choose N n s.t. g divides N, instead of N n If basis state | queries an undefined x i, | drops out of the universe Result: Final state vector has norm in [0,1] Still OK! P(g,N) is exactly polynomial in (g,N); so gs range need not be restricted to [1,n 2/5 ]